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Solution to Mid-term2, CIVL 181
(Spring, 2008)

Problem 1:

T F T T F F T T

Problem 2:
(1)


Let T denote the total time the trip costs, T0 denote the normal time for the trip if no disturbance occurs, D denote the delay time caused by one disturbance, X denote the number of disturbances. Based on the information given in the problem, T0 ~ N(4, 1), D ~ N (0.5, 0.5), and the occurrence of disturbance can be modeled as a poisson with a mean rate of v = 1/50 /km.
If one disturbance occurs,


As T0 and D follow normal distributions, T is also normally distributed. The mean and standard of deviation of T can be calculated as



(T and D are S.I. as assumed in the problem)
The probability that the trip can be completed in 4 hours is


(2) The probabilities of zero, one, two, three, four disturbances are












Accordingly, the probabilities that the trip can be completed in 4 hours when zero, one, two, three and four disturbances occur are,










According to the total probability theorem, the probability the trip can be finished in 4 hours is




(3) Let TD denote the total time caused by all the disturbances.


In this problem,




where D1 to D10 denote the delay time caused by the 10 disturbances, respectively. As D1 to D10 are S.I., the mean and standard deviation of TD are




The mean and standard deviation of T are




As in this problem Dis follow the exponential distribution, TD does not follow any standard probability distribution. However, according to central limit theorem, the probability density function of TD tends to be a normal density function due to the averaging effect. Therefore, it can be assumed here that TD is a normally distributed random variable to simplify the calculation. With this assumption, the total trip time T also follows the normal distribution. In such a case, the probability the torch event will go beyond midnight is



Alternative solution:
Assuming TD follow the Gamma distribution. Let k and v denote the two parameters of the gamma distribution, respectively. Based on the following equations,




k and v can be calculated as




The probability the torch event will go beyond midnight is




Problem 3.
(a)
Let M and F denote the weight of a male and female audience, respectively. Let r denote the percentage of female.
Based on the information given in the problem, the weights of male audiences are fully correlated, and the weights of female audiences are also fully correlated. The means of M and F are 60 kilos and 50 kilos, respectively. The standard deviations of M and F are 15 kilos and 5 kilos, respectively. The mean and standard deviation of r are 0.7 and 0.14, respectively.
The total weight of people in the hall is



(b)
Assuming M, F, and r are S.I., based on first order approximation, the mean and standard